| L(s) = 1 | + 2.05·2-s + 3-s + 2.22·4-s − 5-s + 2.05·6-s + 3.96·7-s + 0.461·8-s + 9-s − 2.05·10-s + 1.88·11-s + 2.22·12-s − 1.65·13-s + 8.15·14-s − 15-s − 3.50·16-s + 1.28·17-s + 2.05·18-s − 5.01·19-s − 2.22·20-s + 3.96·21-s + 3.87·22-s + 6.20·23-s + 0.461·24-s + 25-s − 3.40·26-s + 27-s + 8.82·28-s + ⋯ |
| L(s) = 1 | + 1.45·2-s + 0.577·3-s + 1.11·4-s − 0.447·5-s + 0.839·6-s + 1.50·7-s + 0.163·8-s + 0.333·9-s − 0.649·10-s + 0.569·11-s + 0.642·12-s − 0.459·13-s + 2.18·14-s − 0.258·15-s − 0.875·16-s + 0.310·17-s + 0.484·18-s − 1.15·19-s − 0.497·20-s + 0.866·21-s + 0.827·22-s + 1.29·23-s + 0.0941·24-s + 0.200·25-s − 0.668·26-s + 0.192·27-s + 1.66·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.353369388\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.353369388\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 2.05T + 2T^{2} \) |
| 7 | \( 1 - 3.96T + 7T^{2} \) |
| 11 | \( 1 - 1.88T + 11T^{2} \) |
| 13 | \( 1 + 1.65T + 13T^{2} \) |
| 17 | \( 1 - 1.28T + 17T^{2} \) |
| 19 | \( 1 + 5.01T + 19T^{2} \) |
| 23 | \( 1 - 6.20T + 23T^{2} \) |
| 29 | \( 1 + 1.77T + 29T^{2} \) |
| 31 | \( 1 - 7.05T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 - 6.75T + 41T^{2} \) |
| 43 | \( 1 + 2.39T + 43T^{2} \) |
| 47 | \( 1 - 6.58T + 47T^{2} \) |
| 53 | \( 1 - 3.87T + 53T^{2} \) |
| 59 | \( 1 + 7.70T + 59T^{2} \) |
| 61 | \( 1 - 5.06T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 - 1.20T + 71T^{2} \) |
| 73 | \( 1 - 7.66T + 73T^{2} \) |
| 79 | \( 1 - 6.54T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 8.35T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.908502782885091148541415222815, −7.37110643970525860817809668312, −6.53245304562336618885674154581, −5.78192480535362542047485468267, −4.84387703947989480091684658703, −4.47010873408567340342986128676, −3.92313830592210004820336934548, −2.88037162270421242343849720216, −2.26551082924431520839863340018, −1.10914394729573928039654726403,
1.10914394729573928039654726403, 2.26551082924431520839863340018, 2.88037162270421242343849720216, 3.92313830592210004820336934548, 4.47010873408567340342986128676, 4.84387703947989480091684658703, 5.78192480535362542047485468267, 6.53245304562336618885674154581, 7.37110643970525860817809668312, 7.908502782885091148541415222815
